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The relational complexity of a subgroup  ofSym⁢(Ω)\mathrm{Sym}({\Omega})is a measure of the way in which the orbits of  onΩk\Omega^{k}for various  determine the original action of . Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying betweenPSLn⁢(F)\mathrm{PSL}_{n}(\mathbb{F})andPGLn⁢(F)\mathrm{PGL}_{n}(\mathbb{F}), for an arbitrary field , acting on the set of 1-dimensional subspaces ofFn\mathbb{F}^{n}. We also bound the relational complexity of all groups lying betweenPSLn⁢(q)\mathrm{PSL}_{n}(q)andP⁢Γ⁢Ln⁢(q)\mathrm{P}\Gamma\mathrm{L}_{n}(q), and generalise these results to the action on -spaces form≥1m\geq 1.

The relational complexity of linear groups acting on subspaces